The block exerts a force F on the dart that is proportional to the dart's velocity v and in the opposite direction, that is F = - bv, where b is a constant.

s = mV/2b

Assumptions: 1. the darts finally velocity is zero

2. The force being exerted on the dart by the block is constant and so the dart moves through the block with constant acceleration in the opposite direction. (Newton's second law)

3. Since the acceleration of the dart through the block is constant, then the equations of constant acceleration motion apply to the motion of the dart through the block.

Explanation:

Let a = acceleration of the dart through the block

V = velocity of the dart

m = mass of the dart

Vf = finally velocity of the dart

S = distance traveled by the dart through the block.

From Newton's second law of motion which states that the acceleration of a body is in the same direction as the net force acting on the body and is equal to the force divided by the mass. That is F = ma

Also F = - bv ... (1)

Therefore - bv = ma ... (2)

From the equals of constant acceleration motion, Vf² = V² + 2aS

Vf = 0

0² = V² + 2aS

-2aS = V²

a = - V² / 2S

Substituting this expression for a in

Equation (2) above

-bV = m ( - V²/2S)

On rearranging,

S = mV/2b